1 00:00:00,500 --> 00:00:01,932 [SQUEAKING] 2 00:00:01,932 --> 00:00:03,381 [RUSTLING] 3 00:00:03,381 --> 00:00:04,830 [CLICKING] 4 00:00:11,130 --> 00:00:13,460 SCOTT HUGHES: I'm going to record two lectures today. 5 00:00:13,460 --> 00:00:19,970 These are the final two lectures I will be recording for 8.962. 6 00:00:19,970 --> 00:00:24,170 I actually have notes on an additional two lectures. 7 00:00:24,170 --> 00:00:26,600 I have notes for an additional two lectures. 8 00:00:26,600 --> 00:00:29,540 But those additional two lectures 9 00:00:29,540 --> 00:00:32,180 are somewhat advanced material. 10 00:00:32,180 --> 00:00:36,200 It's sort of fun to go over them in the last week of the course, 11 00:00:36,200 --> 00:00:37,130 for certain students. 12 00:00:37,130 --> 00:00:38,720 It can be a great introduction to some 13 00:00:38,720 --> 00:00:41,180 of the most important topics in modern research. 14 00:00:41,180 --> 00:00:42,788 But I am not going to come into campus 15 00:00:42,788 --> 00:00:45,080 and present those lectures, OK, given everything that's 16 00:00:45,080 --> 00:00:46,760 going on right now. 17 00:00:46,760 --> 00:00:49,250 To be blunt, they are kind of bonus material. 18 00:00:49,250 --> 00:00:51,480 And this isn't the time. 19 00:00:51,480 --> 00:00:54,290 This is not the semester for us to go through our bonuses. 20 00:00:54,290 --> 00:00:57,570 I will make those notes available. 21 00:00:57,570 --> 00:01:00,620 I will be happy to discuss them in a saner moment 22 00:01:00,620 --> 00:01:03,150 with any student who is interested in them. 23 00:01:03,150 --> 00:01:07,100 But this semester, let's just focus on the core material. 24 00:01:07,100 --> 00:01:12,150 So that brings us to the topic of what we are studying today. 25 00:01:12,150 --> 00:01:14,490 Pardon me while I correct my handwriting. 26 00:01:14,490 --> 00:01:17,640 So if I can do a recap of what we discussed last time, 27 00:01:17,640 --> 00:01:23,220 we took a look at a spacetime that has this Schwarzschild 28 00:01:23,220 --> 00:01:25,420 solution written in the Schwarzschild 29 00:01:25,420 --> 00:01:28,080 coordinates everywhere. 30 00:01:28,080 --> 00:01:30,000 In our previous lecture, we looked 31 00:01:30,000 --> 00:01:34,830 at a spacetime that described a fluid body, a spherically 32 00:01:34,830 --> 00:01:36,900 symmetric fluid body. 33 00:01:36,900 --> 00:01:38,230 And it had a surface. 34 00:01:38,230 --> 00:01:40,990 This was the solution that we used for its exterior. 35 00:01:40,990 --> 00:01:45,840 This ends up describing a solution that 36 00:01:45,840 --> 00:01:47,580 has zero stress energy tensor. 37 00:01:47,580 --> 00:01:50,320 So it describes a vacuum situation. 38 00:01:50,320 --> 00:01:50,820 OK? 39 00:01:50,820 --> 00:01:56,370 So if I imagine a spacetime that looks like this everywhere, 40 00:01:56,370 --> 00:02:00,000 well, what I end up finding is that this is a vacuum solution. 41 00:02:00,000 --> 00:02:02,880 It has T-mu-nu equals 0 everywhere. 42 00:02:02,880 --> 00:02:05,820 However, it also has a mass m. 43 00:02:05,820 --> 00:02:07,710 So this is the vacuum solution with mass, 44 00:02:07,710 --> 00:02:11,970 which is, well, that's weird. 45 00:02:11,970 --> 00:02:14,880 We examined some of its curvature properties, 46 00:02:14,880 --> 00:02:18,650 and we found that at r equals 0, there is a tidal singularity, 47 00:02:18,650 --> 00:02:20,220 OK, in invariant quantity that we 48 00:02:20,220 --> 00:02:23,910 constructed from the tidal tensors, blows up at r 49 00:02:23,910 --> 00:02:26,040 equals 0. 50 00:02:26,040 --> 00:02:28,690 R equals 2GM looks a little bit funny. 51 00:02:28,690 --> 00:02:31,920 And it turns out tides are well behaved there. 52 00:02:31,920 --> 00:02:34,687 OK, there's nothing pathological in the spacetime there. 53 00:02:34,687 --> 00:02:36,270 But there is a coordinate singularity. 54 00:02:36,270 --> 00:02:38,700 Our coordinate t is behaving oddly. 55 00:02:38,700 --> 00:02:40,290 And where we left things last time 56 00:02:40,290 --> 00:02:41,850 is I did a little diagnosis of this 57 00:02:41,850 --> 00:02:47,070 by imagining a body that falls into spacetime 58 00:02:47,070 --> 00:02:49,663 from some starting radius R0. 59 00:02:49,663 --> 00:02:51,330 And I looked at the motion of this thing 60 00:02:51,330 --> 00:02:53,310 as a function of time. 61 00:02:53,310 --> 00:02:56,700 What we find is that if we look at the motion of this thing 62 00:02:56,700 --> 00:03:00,090 as a function of the proper time of that in-falling body, 63 00:03:00,090 --> 00:03:04,500 it crosses 2GM in finite proper time. 64 00:03:04,500 --> 00:03:07,530 And shortly afterwards, again, in finite proper time, 65 00:03:07,530 --> 00:03:10,590 it reaches the r equals 0 tidal singularity. 66 00:03:10,590 --> 00:03:14,565 If I look at that same motion as a function 67 00:03:14,565 --> 00:03:17,300 of the coordinate time t, it never 68 00:03:17,300 --> 00:03:22,620 even reaches, never even reaches r equals 2GM. 69 00:03:22,620 --> 00:03:24,180 We found a solution, what you can 70 00:03:24,180 --> 00:03:26,010 see in notes that I've put online 71 00:03:26,010 --> 00:03:29,263 and that are presented in the previous lecture. 72 00:03:29,263 --> 00:03:30,930 But we found a solution in which it just 73 00:03:30,930 --> 00:03:34,350 asymptotically approaches r equals 2GM, 74 00:03:34,350 --> 00:03:39,210 only reaching that radius in the t goes to infinity limit. 75 00:03:39,210 --> 00:03:41,720 These are two starkly different pictures 76 00:03:41,720 --> 00:03:43,790 of the kinematics of this in-falling body. 77 00:03:43,790 --> 00:03:46,120 So the question is, what is going on here? 78 00:03:46,120 --> 00:03:48,260 And as food for thought, I reminded us 79 00:03:48,260 --> 00:03:51,710 that these coordinates, if we sort of look at the way 80 00:03:51,710 --> 00:03:54,860 that spacetime behaves for a very, very, very large M-- 81 00:03:54,860 --> 00:03:56,960 Excuse me, very, very, very large R, 82 00:03:56,960 --> 00:04:01,220 not M, but for a very large R, R much, much greater than 2GM, 83 00:04:01,220 --> 00:04:03,830 this turns into the line element of flat spacetime. 84 00:04:03,830 --> 00:04:06,530 Flat spacetime is what we use in special relativity. 85 00:04:06,530 --> 00:04:08,240 And our time coordinate there, it 86 00:04:08,240 --> 00:04:12,110 is designed using this Einstein synchronization 87 00:04:12,110 --> 00:04:16,279 procedure, which means that the properties of light 88 00:04:16,279 --> 00:04:18,350 as it propagates through spacetime 89 00:04:18,350 --> 00:04:21,740 are built into the coordinate system. 90 00:04:21,740 --> 00:04:23,218 So that suggests what we might want 91 00:04:23,218 --> 00:04:25,760 to start doing in order to try to get some insight as to what 92 00:04:25,760 --> 00:04:30,260 is going on with this weird spacetime is 93 00:04:30,260 --> 00:04:33,770 to think about light as it propagates into spacetime. 94 00:04:33,770 --> 00:04:37,134 Let's think about what happens to radiation as it propagates. 95 00:04:39,860 --> 00:04:53,480 So let's imagine as the body falls that it emits a radio 96 00:04:53,480 --> 00:05:16,630 pulse with a frequency as measured 97 00:05:16,630 --> 00:05:18,070 according to the in falling. 98 00:05:18,070 --> 00:05:19,445 Let's say there's an observer who 99 00:05:19,445 --> 00:05:21,820 is falling in who's got a little radio transmitter that's 100 00:05:21,820 --> 00:05:24,010 beaming this message out. 101 00:05:24,010 --> 00:05:27,820 And according to that observer, this thing is emitted, 102 00:05:27,820 --> 00:05:30,880 the pulse has a frequency omega-- 103 00:05:33,520 --> 00:05:42,440 very far away, we can describe the momentum 104 00:05:42,440 --> 00:05:51,680 of this radio pulse like so. 105 00:05:51,680 --> 00:05:58,440 Imagine the thing is propagating out radially and very, 106 00:05:58,440 --> 00:06:02,510 very far away where the spacetime is approximately 107 00:06:02,510 --> 00:06:03,680 flat. 108 00:06:03,680 --> 00:06:06,920 It's simply the four-momentum that 109 00:06:06,920 --> 00:06:11,180 describes a null geodesic moving in the radial direction, OK? 110 00:06:11,180 --> 00:06:14,000 Minus sign here because with the index in the upstairs position 111 00:06:14,000 --> 00:06:16,190 and this asymptotically flat region, 112 00:06:16,190 --> 00:06:18,030 that would be the energy. 113 00:06:18,030 --> 00:06:20,480 So a couple of facts to bear in mind. 114 00:06:27,770 --> 00:06:38,820 The energy measured by an observer with four-velocity 115 00:06:38,820 --> 00:06:48,090 U is given by, let's call it E sub U, the energy measured 116 00:06:48,090 --> 00:06:57,420 by observer U. This is minus P-dot-U. 117 00:06:57,420 --> 00:07:02,372 We developed this in special relativity. 118 00:07:02,372 --> 00:07:04,830 But remember the way that we use the equivalence principle. 119 00:07:04,830 --> 00:07:06,955 We're gonna use the Einstein equivalence principle. 120 00:07:06,955 --> 00:07:10,470 And any law that holds in a freely falling frame, if I 121 00:07:10,470 --> 00:07:12,570 can write it in a tensorial way that 122 00:07:12,570 --> 00:07:15,300 works in that freely falling frame, it works in any frame. 123 00:07:15,300 --> 00:07:17,820 So this tensorial statement holds true, 124 00:07:17,820 --> 00:07:20,850 even though we are now working in the spacetime that 125 00:07:20,850 --> 00:07:23,640 is distinctly different from special relativity. 126 00:07:23,640 --> 00:07:40,850 We also know that in a time independent spacetime, 127 00:07:40,850 --> 00:07:46,470 the downstairs T component of four momentum is constant. 128 00:07:53,480 --> 00:08:01,130 So what this basically means is if I think about this radio 129 00:08:01,130 --> 00:08:03,290 pulse with its light propagating out 130 00:08:03,290 --> 00:08:08,270 through the spacetime, the value P0 associated with this thing's 131 00:08:08,270 --> 00:08:10,220 four momentum, it's the same everywhere 132 00:08:10,220 --> 00:08:11,868 along its trajectory. 133 00:08:18,570 --> 00:08:25,620 So let's consider a static observer sitting 134 00:08:25,620 --> 00:08:27,099 in the Schwarzschild spacetime. 135 00:08:41,280 --> 00:08:50,130 OK, I'm going to require G-alpha-beta U-alpha 136 00:08:50,130 --> 00:08:52,864 U-beta to be equal to minus 1. 137 00:08:52,864 --> 00:08:58,730 And I will require that this have some timelike piece, 138 00:08:58,730 --> 00:09:00,580 that this observer is static. 139 00:09:00,580 --> 00:09:03,450 So their spatial components of their four-velocity 140 00:09:03,450 --> 00:09:04,880 are equal to 0. 141 00:09:04,880 --> 00:09:07,890 It looks like, by the way-- just pause one moment here. 142 00:09:07,890 --> 00:09:10,110 It looks like the projector is on because of-- 143 00:09:17,127 --> 00:09:18,710 I should probably just leave it as is. 144 00:09:18,710 --> 00:09:20,585 All right, I'm not going to worry about that. 145 00:09:20,585 --> 00:09:21,970 Just leave that as it is. 146 00:09:21,970 --> 00:09:23,690 Sorry, let's go back to what I'm talking about here. 147 00:09:23,690 --> 00:09:25,482 So I have a static observer, so an observer 148 00:09:25,482 --> 00:09:28,330 who is not moving in space. 149 00:09:28,330 --> 00:09:29,830 They're only moving through time. 150 00:09:29,830 --> 00:09:32,823 And I need to normalize their four-velocity. 151 00:09:32,823 --> 00:09:34,990 Bear in mind, this is not a freely-falling observer. 152 00:09:34,990 --> 00:09:38,185 OK, this is an observer who must be accelerated, in some sense. 153 00:09:38,185 --> 00:09:41,140 And there must be some kind of a mechanism that 154 00:09:41,140 --> 00:09:44,410 is allowing this observer to hover at the fixed location 155 00:09:44,410 --> 00:09:47,140 in spacetime where they are at. 156 00:09:47,140 --> 00:09:50,380 So when I put these two constraints together 157 00:09:50,380 --> 00:09:55,670 and I tie it into that spacetime, 158 00:09:55,670 --> 00:10:02,420 what I find is that the timelike component is 1 square root 159 00:10:02,420 --> 00:10:04,550 of 1 minus 2GM over r. 160 00:10:09,730 --> 00:10:17,680 So let's compare the energy that is emitted at radius 161 00:10:17,680 --> 00:10:26,163 r to the energy that is absorbed at some radius big R. So 162 00:10:26,163 --> 00:10:27,080 I'm going to imagine-- 163 00:10:42,146 --> 00:10:44,260 And so the reason I formulate it in this way, what 164 00:10:44,260 --> 00:10:47,560 I want to do now is ask myself, well, 165 00:10:47,560 --> 00:10:52,690 what is the energy that an observer at little r 166 00:10:52,690 --> 00:10:53,500 would measure here? 167 00:11:10,200 --> 00:11:12,190 What is the energy that is observed 168 00:11:12,190 --> 00:11:18,650 by the person at radius capital R? 169 00:11:18,650 --> 00:11:21,500 And now, we know, if I imagine that these are both 170 00:11:21,500 --> 00:11:37,710 being measured by static observers, 171 00:11:37,710 --> 00:11:41,010 because P sub T, P sub 0 or P sub T, 172 00:11:41,010 --> 00:11:59,320 because it is a constant as the light propagates out, 173 00:11:59,320 --> 00:12:02,600 I can take the ratio of these two. 174 00:12:02,600 --> 00:12:11,240 And the energy observed at capital R versus the energy 175 00:12:11,240 --> 00:12:12,350 emitted at little r-- 176 00:12:17,292 --> 00:12:18,000 Hang on a second. 177 00:12:27,950 --> 00:12:29,570 As I was looking over my notes, there 178 00:12:29,570 --> 00:12:35,310 was a result that made no sense because I had a typo. 179 00:12:39,310 --> 00:12:40,810 So for those of you following along, 180 00:12:40,810 --> 00:12:43,660 the four-velocity component should have been a 1 181 00:12:43,660 --> 00:12:45,857 over the square root of that quantity I had earlier. 182 00:12:45,857 --> 00:12:47,440 Suddenly, what I'm about to write down 183 00:12:47,440 --> 00:12:48,400 makes a lot more sense. 184 00:12:52,222 --> 00:12:52,930 So this becomes-- 185 00:13:05,210 --> 00:13:07,130 Let's imagine that the observer is 186 00:13:07,130 --> 00:13:10,730 so far away that they are effectively infinitely 187 00:13:10,730 --> 00:13:11,760 far away. 188 00:13:11,760 --> 00:13:13,910 So the question I'm asking is, if I 189 00:13:13,910 --> 00:13:17,780 imagine that the light pulse is emitted 190 00:13:17,780 --> 00:13:24,020 at some radius little r, what would a very distant observer 191 00:13:24,020 --> 00:13:26,080 measure the energy of that pulse to be? 192 00:13:26,080 --> 00:13:34,390 OK, so my light pulse or radio pulse 193 00:13:34,390 --> 00:13:39,570 is emitted at some finite radius. 194 00:13:39,570 --> 00:13:42,540 And we'll call e infinity, the value that is measured 195 00:13:42,540 --> 00:13:43,860 by a very distant observer. 196 00:13:47,830 --> 00:13:54,380 This becomes square root 1 minus 2GM over r. 197 00:13:54,380 --> 00:13:57,050 Notice that the energy, no matter 198 00:13:57,050 --> 00:14:00,290 how energetic the light is when you emit it, 199 00:14:00,290 --> 00:14:03,350 I described it as being a radio pulse, 200 00:14:03,350 --> 00:14:05,270 but it could be a laser pointer. 201 00:14:05,270 --> 00:14:06,410 It could be ultraviolet. 202 00:14:06,410 --> 00:14:07,835 It could be a gamma ray. 203 00:14:07,835 --> 00:14:09,950 You know, you could just hawk whatever 204 00:14:09,950 --> 00:14:12,665 massively-powerful source of photons you want 205 00:14:12,665 --> 00:14:15,180 and you're pointing it out in the radial direction 206 00:14:15,180 --> 00:14:18,530 and hoping that your distant friend can measure it. 207 00:14:18,530 --> 00:14:22,400 No matter what energy you emit as you are falling in, 208 00:14:22,400 --> 00:14:26,390 as you approach r equals 2GM, the amount 209 00:14:26,390 --> 00:14:31,190 of energy in that beam that reaches a distant observer 210 00:14:31,190 --> 00:14:32,000 goes to 0. 211 00:14:49,660 --> 00:15:04,970 No matter how energetic my pulse of light, 212 00:15:04,970 --> 00:15:16,670 no energy reaches distant observers 213 00:15:16,670 --> 00:15:28,310 as the emitter approaches R equals 2GM. 214 00:15:28,310 --> 00:15:31,460 In a similar way, imagine that as you are falling in 215 00:15:31,460 --> 00:15:33,710 and you've got this little beacon that you are sending 216 00:15:33,710 --> 00:15:37,910 messages out to your distant friends, 217 00:15:37,910 --> 00:15:41,450 imagine you send them out with pulses that are separated 218 00:15:41,450 --> 00:16:04,180 by delta T. So the person in the falling frame 219 00:16:04,180 --> 00:16:06,610 turns his beacon on for a moment every delta 220 00:16:06,610 --> 00:16:08,410 T. Let's say delta T is every second. 221 00:16:21,200 --> 00:16:24,770 The interval between pulses, as measured far away, 222 00:16:24,770 --> 00:16:27,050 one can show using a similar kind of calculation. 223 00:16:52,000 --> 00:16:57,220 You pick up a factor of 1 over square root 1 minus 2GM over r. 224 00:16:57,220 --> 00:17:03,800 This goes to infinity as r goes 2GM. 225 00:17:09,790 --> 00:17:13,099 So remember, the time coordinate was originally 226 00:17:13,099 --> 00:17:17,060 defined by imagining that I can synchronize all of my clocks 227 00:17:17,060 --> 00:17:19,750 using light pulses that are bouncing around. 228 00:17:19,750 --> 00:17:23,540 But what we can kind of see here is that light pulses that 229 00:17:23,540 --> 00:17:27,740 are emitted in the vicinity of r equals 2GM, 230 00:17:27,740 --> 00:17:30,650 they're kind of going to hell, OK? 231 00:17:30,650 --> 00:17:33,120 So if I imagine, let's just say for the sake of argument, 232 00:17:33,120 --> 00:17:35,480 I'm using a green laser pointer as the thing 233 00:17:35,480 --> 00:17:39,110 that I use for my Einstein synchronization procedure. 234 00:17:39,110 --> 00:17:43,760 And suppose that I communicate what time is on the clock 235 00:17:43,760 --> 00:17:46,280 by a modulation of the signal. 236 00:17:46,280 --> 00:17:49,077 So suppose that I modulate it by putting 237 00:17:49,077 --> 00:17:51,410 little spaces on it that tend to be about a second long. 238 00:17:51,410 --> 00:17:53,327 Or let's just say it's like a microsecond long 239 00:17:53,327 --> 00:17:55,850 so I can really pack some information into that. 240 00:17:55,850 --> 00:18:02,920 Well, the laser pointers that are emitting from r equals 2GM, 241 00:18:02,920 --> 00:18:05,350 their signal infinitely redshifts away. 242 00:18:05,350 --> 00:18:06,630 They lose all their energy. 243 00:18:06,630 --> 00:18:07,330 So they're not green. 244 00:18:07,330 --> 00:18:08,710 If they're really close to it, maybe they'll 245 00:18:08,710 --> 00:18:10,335 be red when they get to the next thing. 246 00:18:10,335 --> 00:18:13,660 Instead of being a pulse every microsecond, 247 00:18:13,660 --> 00:18:15,550 it'll be every two microseconds. 248 00:18:15,550 --> 00:18:19,900 And the closer we get to it, the more redshifted 249 00:18:19,900 --> 00:18:22,850 it becomes and the more extended the interval between pulses 250 00:18:22,850 --> 00:18:23,350 becomes. 251 00:18:26,710 --> 00:18:29,890 I'm going to post to the 8.962 website 252 00:18:29,890 --> 00:18:31,570 a set of notes that sort of cleans up 253 00:18:31,570 --> 00:18:33,260 this calculation a little bit, just goes through this 254 00:18:33,260 --> 00:18:34,560 in a little bit more detail. 255 00:18:34,560 --> 00:18:38,260 It was something I wrote in the Spring 2019 256 00:18:38,260 --> 00:18:39,970 semester in response to a good question 257 00:18:39,970 --> 00:18:41,910 that a student had asked me about this. 258 00:18:41,910 --> 00:18:44,260 And I think they do a nice job of just going over 259 00:18:44,260 --> 00:18:45,130 this calculation. 260 00:18:45,130 --> 00:18:46,547 And they give a couple of examples 261 00:18:46,547 --> 00:18:47,770 of the way this is behaving. 262 00:18:47,770 --> 00:18:49,690 The key thing which I want to emphasize 263 00:18:49,690 --> 00:18:57,895 is that as r goes to 2GM, what we see is that-- 264 00:18:57,895 --> 00:18:59,270 well, let's just put it this way. 265 00:18:59,270 --> 00:19:01,030 In fact, let's remove the word "as." 266 00:19:01,030 --> 00:19:11,090 The surface r equals 2GM corresponds 267 00:19:11,090 --> 00:19:12,320 to infinite redshift. 268 00:19:22,170 --> 00:19:24,470 OK, we've talked already a little bit about, 269 00:19:24,470 --> 00:19:28,730 if I have light climbing out of a gravitational field, 270 00:19:28,730 --> 00:19:32,000 the light gets a little bit redder as it climbs out. 271 00:19:32,000 --> 00:19:33,560 Well, at this particular surface, 272 00:19:33,560 --> 00:19:37,830 the sphere of radius 2GM in this spacetime, 273 00:19:37,830 --> 00:19:40,953 if you can get down to there, that redshift 274 00:19:40,953 --> 00:19:41,870 then becomes infinite. 275 00:19:41,870 --> 00:19:46,560 All of the energy is drained out of it as it climbs out. 276 00:19:46,560 --> 00:19:48,800 So what this basically tells us is 277 00:19:48,800 --> 00:19:52,190 that this surface breaks the Einstein synchronization 278 00:19:52,190 --> 00:19:56,120 procedure and it renders that time coordinate bad, 279 00:19:56,120 --> 00:19:59,600 at least if we are concerned about understanding things 280 00:19:59,600 --> 00:20:01,020 right at this surface. 281 00:20:16,530 --> 00:20:20,460 OK, as long as we're concerned with the exterior 282 00:20:20,460 --> 00:20:23,430 of the surface, that's not a problem, OK? 283 00:20:23,430 --> 00:20:25,680 Everything works fine as long as we move away 284 00:20:25,680 --> 00:20:27,120 from this coordinate singularity, 285 00:20:27,120 --> 00:20:29,220 sort of in the same way that many 286 00:20:29,220 --> 00:20:31,800 of the pathologies associated with the spherical 287 00:20:31,800 --> 00:20:33,870 coordinate system here, the north and south pole, 288 00:20:33,870 --> 00:20:36,162 they are fine as long as you're not trying to do things 289 00:20:36,162 --> 00:20:42,850 like measure the longitude angle corresponding to the North 290 00:20:42,850 --> 00:20:43,350 Pole. 291 00:20:43,350 --> 00:20:45,083 So this is not even well-defined. 292 00:20:45,083 --> 00:20:46,500 In the same way, this is basically 293 00:20:46,500 --> 00:20:49,380 telling us that the coordinate T in which we wrote down 294 00:20:49,380 --> 00:20:55,180 this spacetime isn't really well-defined at r equals 2GM. 295 00:20:55,180 --> 00:21:01,930 So what we need to do, if we want to try to get some-- hey, 296 00:21:01,930 --> 00:21:03,340 I'm over here, camera!-- 297 00:21:03,340 --> 00:21:05,650 if we want to try to get a little bit of insight 298 00:21:05,650 --> 00:21:09,794 as to what is going on here, we need a better time coordinate. 299 00:21:21,430 --> 00:21:23,070 So I'm going to talk about a couple. 300 00:21:23,070 --> 00:21:25,710 And the way that we're going to formulate 301 00:21:25,710 --> 00:21:30,210 these is all the pathologies are revealed 302 00:21:30,210 --> 00:21:32,280 when we look at the behavior of light propagating 303 00:21:32,280 --> 00:21:33,047 in this spacetime. 304 00:21:33,047 --> 00:21:34,380 So let's play around with light. 305 00:21:34,380 --> 00:21:37,140 Let's look at null geodesics in this spacetime. 306 00:21:37,140 --> 00:21:40,480 So to begin with, let's stick, for just the next couple 307 00:21:40,480 --> 00:21:43,292 of moments, with the original Schwarzschild coordinates. 308 00:21:54,880 --> 00:21:57,510 OK, so I'm going to look at null geodesics. 309 00:21:57,510 --> 00:22:01,510 So I'm going to set 0, and I'm gonna ask myself, 310 00:22:01,510 --> 00:22:04,440 how does it move through an interval at dt 311 00:22:04,440 --> 00:22:05,910 and an interval of dr? 312 00:22:20,540 --> 00:22:29,570 OK, so we can solve this to find how, 313 00:22:29,570 --> 00:22:32,090 if an object is moving on a radial null geodesic, 314 00:22:32,090 --> 00:22:35,000 how do dt and dr behave? 315 00:22:35,000 --> 00:22:35,983 How does dt dr behave? 316 00:22:35,983 --> 00:22:36,650 Put it that way. 317 00:23:01,150 --> 00:23:04,570 OK, so that's what my dt dr looks like. 318 00:23:04,570 --> 00:23:07,330 Plus, the plus sign corresponds to a solution 319 00:23:07,330 --> 00:23:09,700 that is moving the outward in all direction. 320 00:23:09,700 --> 00:23:13,780 Minus sign is an inward directed null geodesic. 321 00:23:13,780 --> 00:23:17,290 These define what we consider to be the opening angle. 322 00:23:17,290 --> 00:23:23,240 So dt dr defines the opening angle of a light cone. 323 00:23:36,330 --> 00:23:45,620 So if we go to very, very, very large r, 324 00:23:45,620 --> 00:23:47,990 OK, we get dt dr equals 1. 325 00:23:52,750 --> 00:23:56,470 And this corresponds in units in which c equals 1 326 00:23:56,470 --> 00:24:01,570 to light moving on a 45 degree angle in a spacetime diagram. 327 00:24:01,570 --> 00:24:04,675 This is familiar behavior from special relativity. 328 00:24:19,030 --> 00:24:28,505 But as r goes to 2GM, we see dt dr going to 0. 329 00:24:28,505 --> 00:24:29,880 Let's make a sketch and see what, 330 00:24:29,880 --> 00:24:34,020 sort of, dt dr, what the tangent to a null geodesic 331 00:24:34,020 --> 00:24:38,580 looks like in the t-r plane as a function of radius. 332 00:25:20,420 --> 00:25:22,490 So this little dash here means I'm 333 00:25:22,490 --> 00:25:26,000 sort of imagining that I'm going to stretch my r axis so 334 00:25:26,000 --> 00:25:29,030 that, out here, you're in the asymptotically 335 00:25:29,030 --> 00:25:32,332 flat region where things look like special relativity. 336 00:25:40,060 --> 00:25:43,290 So here we are out in this asymptotically flat region. 337 00:25:43,290 --> 00:25:45,540 My outward-going light ray goes off 338 00:25:45,540 --> 00:25:48,040 at 45 degree angle in the r-t plane. 339 00:25:48,040 --> 00:25:52,110 Inward one goes at a 45 degree angle pointing inside. 340 00:25:52,110 --> 00:25:54,320 Down here in the stronger field, it's 341 00:25:54,320 --> 00:25:57,370 going to be a little bit steeper than this. 342 00:25:57,370 --> 00:26:03,360 And so the opening angle of my light cone is closing up. 343 00:26:10,140 --> 00:26:14,550 Here, we'll have closed up a lot more. 344 00:26:19,600 --> 00:26:21,570 Here, it's closed up practically all the way. 345 00:26:21,570 --> 00:26:25,420 Now, as I approach r equals 2GM, both the inward 346 00:26:25,420 --> 00:26:27,040 and the outward direction in these 347 00:26:27,040 --> 00:26:32,160 coordinates go parallel to 2GM. 348 00:26:32,160 --> 00:26:35,580 So you can see the collapse of the light cone 349 00:26:35,580 --> 00:26:47,720 in these coordinates as you approach 350 00:26:47,720 --> 00:26:49,820 this coordinate singularity. 351 00:26:49,820 --> 00:26:52,680 So we need a healthier coordinate system. 352 00:26:52,680 --> 00:26:57,980 One thing that we can do is we can move the pathology out 353 00:26:57,980 --> 00:27:01,070 of our time coordinate and into our radial coordinate 354 00:27:01,070 --> 00:27:02,717 with the following definition. 355 00:27:09,880 --> 00:27:16,480 Suppose you choose a radial coordinate r-star such 356 00:27:16,480 --> 00:27:19,960 that dt equals plus or minus dr-star everywhere. 357 00:27:19,960 --> 00:27:20,560 OK? 358 00:27:20,560 --> 00:27:27,310 So if I replace my horizontal axis with-- 359 00:27:27,310 --> 00:27:30,670 pardon me-- if I replace my horizontal axis 360 00:27:30,670 --> 00:27:35,400 with the r-star, this will be 45 degree angles everywhere. 361 00:27:35,400 --> 00:27:36,190 OK? 362 00:27:36,190 --> 00:27:51,500 But to make this work, what you find is r-star 363 00:27:51,500 --> 00:27:54,360 must look like this. 364 00:27:54,360 --> 00:27:57,620 What you basically see is that this coordinate system 365 00:27:57,620 --> 00:28:00,650 takes r equals 2GM and it moves it 366 00:28:00,650 --> 00:28:03,878 to r-star equals minus infinity. 367 00:28:15,850 --> 00:28:18,780 So the way that this coordinate representation, 368 00:28:18,780 --> 00:28:22,880 that this different radial coordinate, 369 00:28:22,880 --> 00:28:26,190 the way that this makes the light cones always 370 00:28:26,190 --> 00:28:27,373 have 45 degree opening-- 371 00:28:27,373 --> 00:28:29,790 really 90 degrees opening-- it makes the light rays always 372 00:28:29,790 --> 00:28:34,470 go off at 45 degree angles is by essentially constantly 373 00:28:34,470 --> 00:28:37,380 stretching the radial axis so that this guy just 374 00:28:37,380 --> 00:28:39,780 gets stretched out so that it's opening at 90 degrees. 375 00:28:39,780 --> 00:28:41,260 This guy is stretched a little bit less. 376 00:28:41,260 --> 00:28:42,927 This guy is stretched a little bit less. 377 00:28:42,927 --> 00:28:45,280 Basically, not stretched when you're really far away. 378 00:28:45,280 --> 00:28:48,590 But as you approach r equals 2GM, 379 00:28:48,590 --> 00:28:52,050 you're infinitely stretching at these coordinates. 380 00:28:52,050 --> 00:29:11,270 These are known as tortoise coordinates, basically, 381 00:29:11,270 --> 00:29:13,100 because you start walking, taking ever 382 00:29:13,100 --> 00:29:15,650 slower and slower steps. 383 00:29:15,650 --> 00:29:18,140 Even steps, if you imagine you're stepping evenly 384 00:29:18,140 --> 00:29:21,620 in r-star, you're taking ever smaller and smaller steps 385 00:29:21,620 --> 00:29:26,150 in r, as you approach the infinite redshift surface. 386 00:29:29,120 --> 00:29:42,620 So with that tortoise coordinate defined, 387 00:29:42,620 --> 00:29:45,110 you use that as an intermediary to define 388 00:29:45,110 --> 00:29:48,530 a couple of new coordinates for your spacetime that 389 00:29:48,530 --> 00:29:49,970 are adapted to radiation. 390 00:29:55,100 --> 00:30:03,980 So we're going to define v to be t plus r-star. 391 00:30:03,980 --> 00:30:06,320 And the importance of this is that this 392 00:30:06,320 --> 00:30:09,290 is a coordinate that's not hard to convince yourself 393 00:30:09,290 --> 00:30:13,780 this is constant on an in-going radial null ray. 394 00:30:26,240 --> 00:30:31,520 I'm going to define a coordinate U to be t minus r-star. 395 00:30:31,520 --> 00:30:44,884 And this is constant on outgoing radial null rays. 396 00:30:44,884 --> 00:30:45,384 OK? 397 00:30:48,472 --> 00:30:49,930 So this basically means that if I'm 398 00:30:49,930 --> 00:30:55,330 working in this coordinate system, r-star, 399 00:30:55,330 --> 00:30:57,970 if I want to know the behavior of this guy, 400 00:30:57,970 --> 00:30:59,660 well, an outgoing radial null ray, 401 00:30:59,660 --> 00:31:01,660 you just might say, ah, that's the null ray that 402 00:31:01,660 --> 00:31:04,160 has U equals 17. 403 00:31:04,160 --> 00:31:04,660 OK? 404 00:31:04,660 --> 00:31:07,450 And that will then pick out, basically, 405 00:31:07,450 --> 00:31:09,280 a whole sequence of events along which 406 00:31:09,280 --> 00:31:12,008 that null ray has propagated. 407 00:31:15,150 --> 00:31:17,430 And, you know, as you can see, it vastly 408 00:31:17,430 --> 00:31:19,410 simplifies how we describe it. 409 00:31:19,410 --> 00:31:24,030 So once you've defined these two coordinates, 410 00:31:24,030 --> 00:31:26,040 you can rewrite the Schwarzschild spacetime 411 00:31:26,040 --> 00:31:27,090 in terms of them. 412 00:31:27,090 --> 00:31:29,983 It is generally best to choose-- 413 00:31:29,983 --> 00:31:32,400 So we're going to take this another step in just a moment, 414 00:31:32,400 --> 00:31:33,390 but we'll start-- 415 00:31:33,390 --> 00:31:38,770 you choose either V or U, and you replace the Schwarzschild 416 00:31:38,770 --> 00:31:39,270 time. 417 00:31:52,300 --> 00:31:58,730 So let's use V to replace time. 418 00:32:04,600 --> 00:32:07,800 So when you go and you look at what your new coordinate system 419 00:32:07,800 --> 00:32:08,850 looks like-- 420 00:32:08,850 --> 00:32:11,760 and remember, the way you do this is the usual thing, 421 00:32:11,760 --> 00:32:14,820 you're going to make your matrix of partial derivatives 422 00:32:14,820 --> 00:32:16,380 between your old coordinate system. 423 00:32:16,380 --> 00:32:23,400 So your old coordinates are t, r, theta, and phi. 424 00:32:23,400 --> 00:32:28,662 And then they go over to v, r, theta, and phi. 425 00:32:28,662 --> 00:32:29,162 OK? 426 00:32:33,145 --> 00:32:35,520 So you make your matrix of partial derivatives describing 427 00:32:35,520 --> 00:32:36,120 this. 428 00:32:36,120 --> 00:32:39,060 And here's what you find when you 429 00:32:39,060 --> 00:32:45,843 change to the metric in the new representation. 430 00:32:55,520 --> 00:32:58,115 Notice, there's no dr squared term at all. 431 00:32:58,115 --> 00:32:59,330 OK? 432 00:32:59,330 --> 00:33:01,430 We do still see something that's, you know, 433 00:33:01,430 --> 00:33:04,280 at least a coordinate singularity at r equals 2GM. 434 00:33:04,280 --> 00:33:07,430 We haven't quite gotten rid of it entirely here. 435 00:33:07,430 --> 00:33:12,650 But we've definitely mollified the impact 436 00:33:12,650 --> 00:33:14,480 of this coordinate singularity. 437 00:33:14,480 --> 00:33:17,270 So in this coordinate system, you can then 438 00:33:17,270 --> 00:33:22,640 set ds squared equal to 0, and you find two solutions 439 00:33:22,640 --> 00:33:24,290 describing radial null curves. 440 00:33:34,800 --> 00:33:38,600 So dv dr equals 0 for in-going. 441 00:33:41,460 --> 00:33:44,710 And, you know, by definition, these things 442 00:33:44,710 --> 00:33:47,080 are constant on an in-going radial null ray. 443 00:33:47,080 --> 00:33:49,540 And so as you move along it, V remains constant. 444 00:34:02,920 --> 00:34:05,690 OK, and you get something a little bit more complicated 445 00:34:05,690 --> 00:34:06,580 for the outgoing one. 446 00:34:15,710 --> 00:34:19,796 Let's redraw this using my new coordinates, OK? 447 00:34:31,260 --> 00:34:36,996 So I'm going to leave my horizontal axis as r. 448 00:34:36,996 --> 00:34:46,810 So I'm going to make my time axis be v. So out here, 449 00:34:46,810 --> 00:34:49,110 here's my in-going null ray. 450 00:34:53,639 --> 00:34:55,409 And here's my, eh-- let's make that 451 00:34:55,409 --> 00:34:58,170 a little closer to 45 degrees. 452 00:34:58,170 --> 00:35:00,720 Here's my outgoing null ray. 453 00:35:00,720 --> 00:35:02,040 OK? 454 00:35:02,040 --> 00:35:06,920 As I move in to smaller and smaller values of r, 455 00:35:06,920 --> 00:35:11,370 notice in the limit, as r goes to 2GM, 456 00:35:11,370 --> 00:35:13,790 the slope becomes infinite. 457 00:35:13,790 --> 00:35:16,410 OK, so this thing gets steeper. 458 00:35:16,410 --> 00:35:25,050 This one keeps pointing in, so this guy gets steeper, steeper. 459 00:35:25,050 --> 00:35:33,668 Right here, it lies pointing exactly straight up. 460 00:35:33,668 --> 00:35:35,710 What's kind of cool is that in these coordinates, 461 00:35:35,710 --> 00:35:39,750 I can actually look at what it looks like inside this thing. 462 00:35:39,750 --> 00:35:42,490 And so inside, this guy tips over and gets a negative slope 463 00:35:42,490 --> 00:35:43,390 and looks like this. 464 00:35:46,770 --> 00:35:49,160 Now, bear in mind, these two things, 465 00:35:49,160 --> 00:35:51,530 they denote the null rays. 466 00:35:51,530 --> 00:35:59,750 All radial timelike trajectories, 467 00:35:59,750 --> 00:36:01,450 and indeed all timelike trajectories, 468 00:36:01,450 --> 00:36:08,530 not just the radial ones, all timelike trajectories 469 00:36:08,530 --> 00:36:16,690 must follow a world line that is bounded by these two sides. 470 00:36:29,410 --> 00:36:34,050 OK, so if I'm out here, everything in here 471 00:36:34,050 --> 00:36:38,670 describes trajectories that a timelike observer can follow. 472 00:36:38,670 --> 00:36:40,920 Everything in here describes a trajectory 473 00:36:40,920 --> 00:36:44,790 a timelike observer can follow, everything in here, 474 00:36:44,790 --> 00:36:46,230 everything in here. 475 00:36:48,990 --> 00:36:55,200 Notice, when I am at this one here, right at r equals 2GM, 476 00:36:55,200 --> 00:37:00,660 every allowed timelike trajectory points towards r 477 00:37:00,660 --> 00:37:03,770 equals 0. 478 00:37:03,770 --> 00:37:07,820 At best, I can imagine an observer 479 00:37:07,820 --> 00:37:11,030 who's very close to the speed of light who sort of skims 480 00:37:11,030 --> 00:37:13,052 along inside of this thing. 481 00:37:13,052 --> 00:37:14,510 But they are timelike, so they will 482 00:37:14,510 --> 00:37:17,460 have a little bit of a slope that points them inward. 483 00:37:50,560 --> 00:37:51,420 OK. 484 00:37:51,420 --> 00:37:53,700 As you move inside r equals 2GM, it's even more so. 485 00:37:53,700 --> 00:37:55,695 OK, you can't even sort of go parallel to the r 486 00:37:55,695 --> 00:37:56,580 equals 2GM line. 487 00:37:56,580 --> 00:37:58,230 They all point towards this thing. 488 00:38:01,290 --> 00:38:04,320 Once you get to r equals 2GM, all trajectories, 489 00:38:04,320 --> 00:38:07,740 as they move to the future, must move to smaller radius. 490 00:38:33,360 --> 00:38:36,690 What this tells us, in particular, 491 00:38:36,690 --> 00:38:43,050 is that once you have reached r equals 2GM, 492 00:38:43,050 --> 00:38:44,920 you are never coming back. 493 00:39:10,830 --> 00:39:13,140 All allowed trajectories, everything 494 00:39:13,140 --> 00:39:15,480 that is permissible by the laws of physics, 495 00:39:15,480 --> 00:39:17,550 moves along a trajectory that points towards r 496 00:39:17,550 --> 00:39:20,940 equals 0 once you hit that r equals 2GM line. 497 00:39:20,940 --> 00:39:25,020 Because of this, this surface r, this surface 498 00:39:25,020 --> 00:39:27,510 of infinite redshift is given the name-- 499 00:39:33,888 --> 00:39:35,680 Let me come back to the point I was making. 500 00:39:35,680 --> 00:39:39,420 So the surface r equal 2GM, nothing that crosses 501 00:39:39,420 --> 00:39:41,550 it is ever going to come back. 502 00:39:41,550 --> 00:39:47,530 This surface of infinite redshift, 503 00:39:47,530 --> 00:39:49,700 we call an event horizon. 504 00:40:00,080 --> 00:40:02,290 OK? 505 00:40:02,290 --> 00:40:05,725 No events that are on the other side of r 506 00:40:05,725 --> 00:40:10,030 equals 2GM can have any causal influence on events 507 00:40:10,030 --> 00:40:10,700 on the outside. 508 00:40:42,000 --> 00:40:44,910 If you have a spacetime with an event horizon like this 509 00:40:44,910 --> 00:40:47,020 and this one that we are talking about right here, 510 00:40:47,020 --> 00:40:49,170 I'm going to talk at the end of my final lecture 511 00:40:49,170 --> 00:40:51,435 that I record about-- 512 00:40:51,435 --> 00:40:53,310 Maybe, actually, I'm gonna do it in this one. 513 00:40:53,310 --> 00:40:54,030 Yeah, I am. 514 00:40:54,030 --> 00:40:55,770 So at the end of this lecture that I'm recording, 515 00:40:55,770 --> 00:40:57,812 I'm going to go over a couple of other spacetimes 516 00:40:57,812 --> 00:40:59,590 that have this structure. 517 00:40:59,590 --> 00:41:02,790 Such spacetimes are called-- 518 00:41:09,970 --> 00:41:32,080 wait for it-- such spacetimes are called black holes. 519 00:41:32,080 --> 00:41:36,040 They are black because light cannot get out of them. 520 00:41:36,040 --> 00:41:39,010 And they are holes because you just jump into 'em, 521 00:41:39,010 --> 00:41:42,180 and you ain't never coming out. 522 00:41:42,180 --> 00:41:43,720 So let me pause for just one moment. 523 00:41:43,720 --> 00:41:46,720 I want to send a quick note to the ODL person who 524 00:41:46,720 --> 00:41:49,916 is helping me out here. 525 00:41:49,916 --> 00:41:52,030 I just want to let her know that this lecture may 526 00:41:52,030 --> 00:42:00,033 run a tiny bit long, since I spent a moment 527 00:42:00,033 --> 00:42:02,200 chatting with a police officer who checked in on me. 528 00:42:30,968 --> 00:42:34,920 OK, if you're watching, Elaine, hi! 529 00:42:34,920 --> 00:42:36,860 So I just sent you a quick note, letting 530 00:42:36,860 --> 00:42:41,150 you know that I'm likely to run a little bit long. 531 00:42:41,150 --> 00:42:43,530 All right, let's get back to black holes. 532 00:42:43,530 --> 00:42:49,010 So to sort of call out some of the structure 533 00:42:49,010 --> 00:42:53,090 of this spacetime, I want to spend just a few minutes 534 00:42:53,090 --> 00:42:58,160 talking about one final coordinate transformation that 535 00:42:58,160 --> 00:43:02,880 is very useful, but looks really weird. 536 00:43:02,880 --> 00:43:06,170 So just bear with me as I go through this-- 537 00:43:06,170 --> 00:43:09,560 a very useful, but unquestionably somewhat 538 00:43:09,560 --> 00:43:11,613 obtuse coordinate transformation. 539 00:43:20,500 --> 00:43:23,288 What I'm going to do is I'm going 540 00:43:23,288 --> 00:43:24,580 to define a coordinate v-prime. 541 00:43:32,490 --> 00:43:35,260 This is given by taking the exponential 542 00:43:35,260 --> 00:43:45,010 of the in-going coordinate time V, normalized for GM. 543 00:43:45,010 --> 00:43:49,570 U prime will be the exponent of U, 544 00:43:49,570 --> 00:43:53,020 a time that works well for the outgoing coordinate system 545 00:43:53,020 --> 00:43:55,090 divided by 4GM. 546 00:43:55,090 --> 00:44:00,420 I'm then going to define capital T 547 00:44:00,420 --> 00:44:08,720 to be 1/2 V-prime plus U-prime, capital 548 00:44:08,720 --> 00:44:13,400 R to be 1/2 V-prime minus U-prime. 549 00:44:23,500 --> 00:44:27,770 It's then simple to show, where simple 550 00:44:27,770 --> 00:44:30,440 is professor speak for "a student can probably 551 00:44:30,440 --> 00:44:31,895 do it in an hour or so." 552 00:44:31,895 --> 00:44:34,670 It's sort of tedious, but straightforward, 553 00:44:34,670 --> 00:44:36,620 just hooking together lots of definitions 554 00:44:36,620 --> 00:44:41,660 and slogging through a couple of identities. 555 00:44:50,160 --> 00:44:56,400 It's simple to show that capital T relates 556 00:44:56,400 --> 00:45:03,010 to Schwarzschild time T and Schwarzschild radius r like so. 557 00:45:03,010 --> 00:45:03,885 There's two branches. 558 00:45:42,790 --> 00:45:47,580 OK, so this is how one relates capital T and capital 559 00:45:47,580 --> 00:45:54,460 R to Schwarzschild t and Schwarzschild r in the region 560 00:45:54,460 --> 00:45:57,490 r greater than or equal to 2GM. 561 00:46:03,150 --> 00:46:48,990 You find a somewhat different solution 562 00:46:48,990 --> 00:46:51,768 in the region r less than 2GM. 563 00:46:51,768 --> 00:46:53,310 So if you're looking at this and kind 564 00:46:53,310 --> 00:46:56,250 of going, "what the hell are you talking about here," 565 00:46:56,250 --> 00:46:56,810 that's fine. 566 00:47:03,446 --> 00:47:06,496 Let me just write down two more relationships. 567 00:47:11,460 --> 00:47:13,560 And then I'll describe what this is good for. 568 00:47:26,110 --> 00:47:28,660 So a particularly clean inversion 569 00:47:28,660 --> 00:47:48,460 between TR and the original Schwarzschild tr, 570 00:47:48,460 --> 00:47:56,720 both the r greater than 2GM and r less than 2GM branches 571 00:47:56,720 --> 00:47:58,055 can be subsumed into this. 572 00:48:07,570 --> 00:48:16,360 And you find T over R looks like the hyperbolic tangent 573 00:48:16,360 --> 00:48:25,920 T over 4GM when you're in the exterior 574 00:48:25,920 --> 00:48:34,060 and the hyperbolic cotangent in the interior. 575 00:48:34,060 --> 00:48:38,780 So these rather bizarre-looking coordinates, 576 00:48:38,780 --> 00:48:52,806 these are known as Kruskal-Szekeres coordinates. 577 00:48:59,663 --> 00:49:01,080 I'll just leave that down like so. 578 00:49:04,740 --> 00:49:06,987 So when one goes into this, I'm not going to deny it, 579 00:49:06,987 --> 00:49:08,820 this is a bizarre looking coordinate system. 580 00:49:08,820 --> 00:49:11,490 OK, but it's got several features 581 00:49:11,490 --> 00:49:15,960 that make it very useful for understanding 582 00:49:15,960 --> 00:49:18,260 what is going on physically in this spacetime. 583 00:49:44,110 --> 00:49:46,990 So first, if you rewrite your metric 584 00:49:46,990 --> 00:49:53,050 in terms of capital T and capital R, what you get 585 00:49:53,050 --> 00:49:55,680 is a form that has no singularities. 586 00:49:55,680 --> 00:49:59,620 It's well-behaved everywhere, except at r equals 0. 587 00:50:10,100 --> 00:50:12,530 So you do get a singularity there, 588 00:50:12,530 --> 00:50:14,810 things blow up as r goes to 0. 589 00:50:22,840 --> 00:50:24,903 There's no other coordinate pathologies. 590 00:50:27,748 --> 00:50:29,540 And then you get sort of an angular sector. 591 00:50:29,540 --> 00:50:32,480 It's actually cleanest in terms of the Schwarzschild radius r, 592 00:50:32,480 --> 00:50:34,330 so we'll leave it in terms of that. 593 00:50:34,330 --> 00:50:36,400 One thing which is nice is notice 594 00:50:36,400 --> 00:50:51,050 that radial null geodesics, they simply 595 00:50:51,050 --> 00:50:59,085 obey dt equals plus or minus dr everywhere, OK? 596 00:50:59,085 --> 00:51:01,460 The only place where you run into a little bit of problem 597 00:51:01,460 --> 00:51:05,360 is as r goes to 0. 598 00:51:05,360 --> 00:51:07,910 And that's special, OK? 599 00:51:07,910 --> 00:51:10,910 So I got that just by setting ds squared equal to 0. 600 00:51:10,910 --> 00:51:13,280 It's radial, so my D-omega goes to 0. 601 00:51:13,280 --> 00:51:19,080 And then just dt equals plus or minus dr. 602 00:51:19,080 --> 00:51:20,840 So that's really nice, OK? 603 00:51:20,840 --> 00:51:22,960 I'm gonna make a sketch in just a moment. 604 00:51:22,960 --> 00:51:26,940 And the fact that I know light always moves along 605 00:51:26,940 --> 00:51:29,220 45 degree lines in this coordinate system 606 00:51:29,220 --> 00:51:32,310 is going to help me to understand the causal structure 607 00:51:32,310 --> 00:51:33,784 of this spacetime. 608 00:51:40,210 --> 00:51:43,670 The causal structure is what I mean by which events 609 00:51:43,670 --> 00:51:45,170 can influence other events. 610 00:51:51,840 --> 00:51:54,450 What can exert a causal influence on what? 611 00:52:03,410 --> 00:52:06,890 So as I move on, I'm going to make a sketch in just a moment, 612 00:52:06,890 --> 00:52:10,010 I want to highlight a couple of behaviors 613 00:52:10,010 --> 00:52:12,920 that we see that are really sort of called out 614 00:52:12,920 --> 00:52:16,805 in this mapping between the two coordinate systems. 615 00:52:20,480 --> 00:52:29,360 So notice that a surface of constant Schwarzschild 616 00:52:29,360 --> 00:52:37,780 radius, constant r forms a hyperbola 617 00:52:37,780 --> 00:52:40,311 in the Kruskal-Szekeres coordinates. 618 00:53:03,410 --> 00:53:09,020 Notice that surfaces of constant time, 619 00:53:09,020 --> 00:53:11,330 they form lines in that they form 620 00:53:11,330 --> 00:53:14,870 lines of slope t over r equal to some constant. 621 00:53:32,020 --> 00:53:39,370 So they are lines with t over r equaling, on the exterior, 622 00:53:39,370 --> 00:53:41,510 let's just focus on the exterior, 623 00:53:41,510 --> 00:53:45,530 they have a slope that's given by the hyperbolic tangent of t 624 00:53:45,530 --> 00:53:46,030 over 4GM. 625 00:53:53,720 --> 00:53:56,450 On the interior-- replaced with cotangent, 626 00:53:56,450 --> 00:53:57,997 hyperbolic cotangent. 627 00:54:03,700 --> 00:54:06,200 The last thing which I'd like to note before I make a sketch 628 00:54:06,200 --> 00:54:10,520 here is let's look at the special surface 629 00:54:10,520 --> 00:54:12,770 of infinite redshift, this event horizon. 630 00:54:16,790 --> 00:54:35,150 So if I plug in r equals 2GM, plug this in over here, 631 00:54:35,150 --> 00:54:38,810 I get t squared minus r squared equals 0. 632 00:54:38,810 --> 00:54:41,300 This is the asymptotic limit to those hyperbolae. 633 00:54:41,300 --> 00:54:50,260 They just become lines t equals plus or minus r. 634 00:54:50,260 --> 00:54:53,980 So the event horizon in this coordinate representation 635 00:54:53,980 --> 00:54:57,790 is just going to be a pair of lines crossing 636 00:54:57,790 --> 00:55:00,470 in the origin of these coordinate systems, 637 00:55:00,470 --> 00:55:02,530 OK, a pair of 45 degree lines crossing 638 00:55:02,530 --> 00:55:05,260 into this coordinate system. 639 00:55:05,260 --> 00:55:10,690 Notice, also, that t equals plus or minus r, 640 00:55:10,690 --> 00:55:25,260 this corresponds to Schwarzschild t going to 641 00:55:25,260 --> 00:55:28,720 plus or minus infinity. 642 00:55:28,720 --> 00:55:34,260 So this, indeed, is a weird, singular limit 643 00:55:34,260 --> 00:55:37,560 of the Schwarzschild time coordinate. 644 00:55:37,560 --> 00:55:42,030 So you can find much prettier versions of this figure 645 00:55:42,030 --> 00:55:46,800 than the one I'm about to attempt to sketch. 646 00:55:46,800 --> 00:55:49,300 Let's see what I can do with this. 647 00:55:49,300 --> 00:55:56,630 So horizontal will be the Kruskal-Szekeres coordinate 648 00:55:56,630 --> 00:56:00,620 r, vertical will be the coordinate t. 649 00:56:16,190 --> 00:56:28,070 Here is the event horizon, r equals t or little r 650 00:56:28,070 --> 00:56:30,960 equals 2GM. 651 00:56:30,960 --> 00:56:44,870 Some different surface of r equal to some constant value 652 00:56:44,870 --> 00:56:50,500 greater than 2GM will live on a hyperbola like so. 653 00:56:57,810 --> 00:57:02,630 Some value of r equals constant, but less than 2GM, 654 00:57:02,630 --> 00:57:04,500 lies on a hyperbola like so. 655 00:57:14,760 --> 00:57:17,550 In particular, there is one special hyperbole 656 00:57:17,550 --> 00:57:19,380 corresponding to r equals 0. 657 00:57:22,390 --> 00:57:30,530 And this is where my artistry is going to truly fail me. 658 00:57:30,530 --> 00:57:33,442 This is an infinite tidal singularity. 659 00:57:36,610 --> 00:57:38,110 Now, the thing which is particularly 660 00:57:38,110 --> 00:57:42,010 useful about this particular coordinate system 661 00:57:42,010 --> 00:57:47,140 is, remember, light always moves in the capital R, 662 00:57:47,140 --> 00:57:49,660 capital T coordinates. 663 00:57:49,660 --> 00:57:56,650 It always moves on lines that go dt equals plus or minus dr. So 664 00:57:56,650 --> 00:58:01,960 what you can see is that imagine I start here 665 00:58:01,960 --> 00:58:05,332 and I send out a little light pulse, 666 00:58:05,332 --> 00:58:10,790 OK, a radially outgoing light pulse, it will always 667 00:58:10,790 --> 00:58:17,090 go away and go to larger and larger values of r, just sort 668 00:58:17,090 --> 00:58:20,840 of constantly moves along this particular trajectory. 669 00:58:20,840 --> 00:58:23,960 Let me write out again what I'm doing here. 670 00:58:23,960 --> 00:58:36,670 So a radial outgoing light ray, it will follow dt equals dr. 671 00:58:36,670 --> 00:58:40,840 But notice that this line goes parallel to the event horizon. 672 00:58:40,840 --> 00:58:43,720 If I am on the inside of this guy 673 00:58:43,720 --> 00:58:48,310 and I try to make a light pulse that goes outside, 674 00:58:48,310 --> 00:58:51,430 points in the radial direction, all it does 675 00:58:51,430 --> 00:58:53,470 is, in these coordinates, it moves parallel 676 00:58:53,470 --> 00:58:54,430 to the event horizon. 677 00:58:54,430 --> 00:58:56,780 It can never cross it. 678 00:58:56,780 --> 00:58:59,560 And in fact, because this is a hyperbola, 679 00:58:59,560 --> 00:59:01,480 one finds that even though you have 680 00:59:01,480 --> 00:59:05,530 tried to make this guy as outgoing as outgoing can be, 681 00:59:05,530 --> 00:59:13,920 it will eventually intersect the r equals 0 tidal singularity. 682 00:59:13,920 --> 00:59:17,260 Since I cannot have any event, here, 683 00:59:17,260 --> 00:59:20,570 that communicates with any event on the other side of this, 684 00:59:20,570 --> 00:59:47,170 this region, everything at r less than 2GM, 685 00:59:47,170 --> 00:59:52,060 these will be causally disconnected-- 686 00:59:52,060 --> 00:59:54,460 not "casually," pardon me. 687 01:00:04,520 --> 01:00:13,010 They are causally disconnected from the world 688 01:00:13,010 --> 01:00:14,880 that lies outside of r is 2GM. 689 01:00:18,740 --> 01:00:21,340 So in my notes and in Carroll's textbook, 690 01:00:21,340 --> 01:00:23,080 there is another coordinate system 691 01:00:23,080 --> 01:00:25,920 that you can do which essentially takes 692 01:00:25,920 --> 01:00:27,660 points that are infinitely far away 693 01:00:27,660 --> 01:00:32,160 and brings them into a finite coordinate location. 694 01:00:32,160 --> 01:00:35,802 And that final thing, it puts it in what 695 01:00:35,802 --> 01:00:37,635 are called Penrose coordinates and it allows 696 01:00:37,635 --> 01:00:40,560 you make what's called the Penrose diagram, which 697 01:00:40,560 --> 01:00:46,200 displays, in a very simple way, how different events are either 698 01:00:46,200 --> 01:00:50,610 causally connected or causally disconnected from the others. 699 01:00:50,610 --> 01:00:54,150 Fairly advanced stuff, not important, but many of you 700 01:00:54,150 --> 01:00:55,530 may find it interesting. 701 01:00:55,530 --> 01:00:58,110 Happy to talk further, once we all 702 01:00:58,110 --> 01:01:00,825 have the bandwidth to have those kinds of conversations. 703 01:01:06,880 --> 01:01:08,218 So let me summarize. 704 01:01:27,250 --> 01:01:31,440 So the summary is that this spacetime, so earlier we 705 01:01:31,440 --> 01:01:34,740 were looking at the spacetime of a spherically symmetric fluid 706 01:01:34,740 --> 01:01:36,390 object. 707 01:01:36,390 --> 01:01:39,480 We found a particularly clean form 708 01:01:39,480 --> 01:01:42,480 for the exterior of that object, where it was a vacuum solution. 709 01:02:01,900 --> 01:02:06,490 If we imagine a spacetime that has this everywhere, 710 01:02:06,490 --> 01:02:16,220 then we get this solution that we call a black hole. 711 01:02:16,220 --> 01:02:18,520 So I emphasize this is vacuum everywhere, 712 01:02:18,520 --> 01:02:23,890 but sort of the analysis kind of goes to hell at r equals 0. 713 01:02:23,890 --> 01:02:26,890 So there are some singular field equations there trying 714 01:02:26,890 --> 01:02:28,600 to describe the stress energy. 715 01:02:28,600 --> 01:02:31,810 Its behavior as you approach there, 716 01:02:31,810 --> 01:02:36,460 let's just say we're not quite sure what's going on. 717 01:02:36,460 --> 01:02:40,810 In this coordinate system, we see weird things 718 01:02:40,810 --> 01:02:49,730 happening as we approach r equals 2GM. 719 01:02:49,730 --> 01:02:51,500 This is simply a coordinate singularity. 720 01:02:51,500 --> 01:02:55,850 There's really nothing going bad with the physics here. 721 01:02:55,850 --> 01:02:59,060 But our attempts to use a time coordinate 722 01:02:59,060 --> 01:03:02,510 that's based on, essentially, the way light 723 01:03:02,510 --> 01:03:05,270 moves in empty space, it's failing in this region. 724 01:03:05,270 --> 01:03:08,600 And all of this lecture is about uncovering this and seeing 725 01:03:08,600 --> 01:03:13,700 that, in fact, this is a surface of infinite redshift beyond 726 01:03:13,700 --> 01:03:17,000 which things cannot communicate. 727 01:03:17,000 --> 01:03:24,290 So this is one of the big discoveries that came out 728 01:03:24,290 --> 01:03:27,200 of general relativity, OK, this creature we 729 01:03:27,200 --> 01:03:29,860 call the black hole. 730 01:03:29,860 --> 01:03:33,140 It is not the only solution of the Einstein field equations 731 01:03:33,140 --> 01:03:35,600 that we call a black hole. 732 01:03:35,600 --> 01:03:38,690 Let me talk briefly about two others. 733 01:03:54,010 --> 01:03:58,960 So another one has a spacetime that looks like this. 734 01:04:28,410 --> 01:04:30,030 So where did this come from? 735 01:04:30,030 --> 01:04:32,240 Well, suppose I bung this through the Einstein field 736 01:04:32,240 --> 01:04:37,520 equations, what I find is that it comes from a non-zero stress 737 01:04:37,520 --> 01:04:38,300 energy tensor. 738 01:04:38,300 --> 01:04:44,660 In fact, it's a stress energy tensor that looks like this. 739 01:04:44,660 --> 01:04:45,644 Pardon me. 740 01:05:14,220 --> 01:05:18,990 This is a stress energy tensor of a Coulomb electric field 741 01:05:18,990 --> 01:05:20,130 with total charge q. 742 01:05:28,720 --> 01:05:31,480 This represents a charged black hole. 743 01:06:09,540 --> 01:06:11,790 It turns out if you analyze this thing carefully, 744 01:06:11,790 --> 01:06:13,980 you find it has an event horizon. 745 01:06:19,570 --> 01:06:30,140 It turns out to be located at G quantity that 746 01:06:30,140 --> 01:06:34,110 involves the square root of m squared minus q squared. 747 01:06:34,110 --> 01:06:37,820 Now, if q-- don't even ask me what the units are that this is 748 01:06:37,820 --> 01:06:40,490 being measured in, they're pretty goofy units-- 749 01:06:40,490 --> 01:06:43,580 but if, in these units, the magnitude of q 750 01:06:43,580 --> 01:06:47,000 is greater than m, there is no horizon. 751 01:06:47,000 --> 01:06:52,340 There is still, however, an infinite tidal singularity at r 752 01:06:52,340 --> 01:06:53,780 equals 0. 753 01:06:53,780 --> 01:06:56,540 So such a solution would give us what is 754 01:06:56,540 --> 01:06:58,530 known as a naked singularity. 755 01:07:01,290 --> 01:07:04,530 I have a few comments and I have a couple notes on these. 756 01:07:04,530 --> 01:07:06,710 But they're not as important as other things 757 01:07:06,710 --> 01:07:07,870 I'd like to talk about. 758 01:07:07,870 --> 01:07:10,570 You might wonder, where does that r horizon actually 759 01:07:10,570 --> 01:07:11,458 come from? 760 01:07:11,458 --> 01:07:13,750 OK, that is what you get when you find the route, where 761 01:07:13,750 --> 01:07:17,793 you look for the place where the metric function vanishes. 762 01:07:17,793 --> 01:07:19,960 OK, you notice there's only one metric function that 763 01:07:19,960 --> 01:07:22,960 appears in there, 1 minus 2GM over r plus q 764 01:07:22,960 --> 01:07:24,800 squared over r squared. 765 01:07:24,800 --> 01:07:27,490 So in general, if you have what's 766 01:07:27,490 --> 01:07:31,330 known as a stationary spacetime, you can find 767 01:07:31,330 --> 01:07:36,440 coordinates such that surfaces of constant r 768 01:07:36,440 --> 01:07:40,110 are spacelike surfaces. 769 01:07:40,110 --> 01:07:43,590 If you look for the place where a surface of constant r 770 01:07:43,590 --> 01:07:49,290 makes a transition from being a spacelike surface to being 771 01:07:49,290 --> 01:07:55,980 a null surface, that tells you that you 772 01:07:55,980 --> 01:08:00,470 have located an event horizon. 773 01:08:32,580 --> 01:08:34,640 OK, this is discussed in a little bit more detail 774 01:08:34,640 --> 01:08:35,359 in my notes. 775 01:08:35,359 --> 01:08:36,950 There's also some very nice discussion 776 01:08:36,950 --> 01:08:40,040 in Carroll's textbook. 777 01:08:40,040 --> 01:08:41,840 What it boils down to is that if you 778 01:08:41,840 --> 01:08:45,680 can find a radial coordinate that allows you to do this, 779 01:08:45,680 --> 01:08:48,649 then the condition G upstairs r upstairs r 780 01:08:48,649 --> 01:08:52,514 equals 0 defines your event horizon, OK? 781 01:08:52,514 --> 01:08:55,010 It just so happens that it's also 782 01:08:55,010 --> 01:08:57,800 equal to G downstairs t downstairs t equals 0, 783 01:08:57,800 --> 01:08:59,899 in this case and in the Schwarzschild case. 784 01:08:59,899 --> 01:09:02,840 But it's not like that for all black holes 785 01:09:02,840 --> 01:09:04,310 that you can write down. 786 01:09:04,310 --> 01:09:07,760 In particular, let me write down the final black hole 787 01:09:07,760 --> 01:09:11,479 spacetime I want to discuss in this lecture. 788 01:09:15,564 --> 01:09:17,439 This is gonna take a minute, so bear with me. 789 01:10:47,090 --> 01:10:53,500 OK, so in this spacetime, the symbol delta I've written here, 790 01:10:53,500 --> 01:11:00,430 this is r-squared minus 2GM r plus a squared. 791 01:11:00,430 --> 01:11:07,093 Rho squared is r squared plus a squared cosine 792 01:11:07,093 --> 01:11:08,260 of the square root of theta. 793 01:11:11,040 --> 01:11:14,940 This thing turns out, if you compute 794 01:11:14,940 --> 01:11:17,850 the inverse metric components, you 795 01:11:17,850 --> 01:11:23,760 find that g upstairs r upstairs r is proportional to delta. 796 01:11:23,760 --> 01:11:31,230 And so there is a horizon where delta equals 0. 797 01:11:46,660 --> 01:11:47,160 Sorry. 798 01:11:59,547 --> 01:12:02,130 That turns out to be located at a radius that looks like this. 799 01:12:05,760 --> 01:12:13,330 This represents the spacetime of a spinning black hole. 800 01:12:13,330 --> 01:12:18,870 It was discovered by Roy Kerr, a mathematician from New Zealand. 801 01:12:21,570 --> 01:12:25,100 I think this was actually a big part of his PhD work. 802 01:12:35,570 --> 01:12:41,590 So it is known as a Kerr black hole. 803 01:12:41,590 --> 01:12:49,780 The parameter a is related to the angular momentum, the spin 804 01:12:49,780 --> 01:12:53,720 angular momentum of the black hole in the units 805 01:12:53,720 --> 01:12:56,090 that we measure these, normalized to the mass. 806 01:12:58,790 --> 01:13:09,310 So notice that in order for this to actually have a horizon, 807 01:13:09,310 --> 01:13:13,900 you need that a to be less than or equal to GM. 808 01:13:13,900 --> 01:13:16,990 If it saturates that bound, then you 809 01:13:16,990 --> 01:13:19,750 get what's known as a maximal black hole. 810 01:13:19,750 --> 01:13:21,550 So it has a couple of noteworthy features. 811 01:13:31,180 --> 01:13:35,570 First, it is not spherically symmetric. 812 01:13:44,070 --> 01:13:46,530 If it were spherically symmetric, 813 01:13:46,530 --> 01:13:50,400 we could write the d theta squared d phi squared piece, 814 01:13:50,400 --> 01:13:53,130 we could find some radial coordinate such 815 01:13:53,130 --> 01:14:06,190 that there was some simple radius 816 01:14:06,190 --> 01:14:13,840 such that G theta theta was simply sine 817 01:14:13,840 --> 01:14:16,420 squared G theta theta. 818 01:14:16,420 --> 01:14:19,870 This is the condition that defines spherical symmetry. 819 01:14:19,870 --> 01:14:25,280 And there is no angle-independent radial 820 01:14:25,280 --> 01:14:28,040 coordinate that allows you to do that. 821 01:14:28,040 --> 01:14:37,730 Notice also that there is a connection 822 01:14:37,730 --> 01:14:40,260 in this coordinate system between t and phi. 823 01:14:46,230 --> 01:14:58,070 Gt phi is equal to minus 2 GM a r sine squared theta over rho 824 01:14:58,070 --> 01:14:59,230 squared. 825 01:14:59,230 --> 01:15:00,770 Why minus 2 and not minus 4? 826 01:15:00,770 --> 01:15:06,510 Well, remember what I have there is Gt phi dt d phi plus G phi 827 01:15:06,510 --> 01:15:10,150 t d phi dt. 828 01:15:10,150 --> 01:15:15,040 One can show that this term, it reflects the kind 829 01:15:15,040 --> 01:15:19,180 of physics in which the spin of the black hole 830 01:15:19,180 --> 01:15:22,900 introduces a spinning, almost magnetic-like element 831 01:15:22,900 --> 01:15:24,760 to gravitation. 832 01:15:24,760 --> 01:15:26,810 If you guys do the homework assignment 833 01:15:26,810 --> 01:15:33,040 I have assigned in which you compute the linearized effect 834 01:15:33,040 --> 01:15:36,350 on a spacetime of a spinning body, 835 01:15:36,350 --> 01:15:37,790 you'll get a flavor of this, OK? 836 01:15:37,790 --> 01:15:41,060 This ends up giving you, that calculation gives you 837 01:15:41,060 --> 01:15:44,360 a similar term in the spacetime which further analysis 838 01:15:44,360 --> 01:15:46,370 shows leads to bodies. 839 01:15:46,370 --> 01:15:49,850 Essentially, what you find is that if you have an orbit that 840 01:15:49,850 --> 01:15:53,240 goes in the same sense as the body's 841 01:15:53,240 --> 01:15:56,380 spin versus an orbit that goes in the opposite sense 842 01:15:56,380 --> 01:15:58,430 of the body's spin, there's a splitting 843 01:15:58,430 --> 01:16:00,390 in the orbit's properties due to that. 844 01:16:00,390 --> 01:16:00,890 OK? 845 01:16:00,890 --> 01:16:02,720 So it breaks the symmetry between what 846 01:16:02,720 --> 01:16:07,400 we call a prograde orbit and a retrograde orbit. 847 01:16:07,400 --> 01:16:11,660 This is one of the most important solutions 848 01:16:11,660 --> 01:16:15,260 that we know of in general relativity because of a result 849 01:16:15,260 --> 01:16:16,520 that I'm going to discuss now. 850 01:16:20,970 --> 01:16:23,540 Oh, first of all, I should mention that, in fact, you 851 01:16:23,540 --> 01:16:28,520 can combine charge with spin. 852 01:16:28,520 --> 01:16:32,390 I'm not going to write down the result because it's just 853 01:16:32,390 --> 01:16:33,060 kind of messy. 854 01:16:33,060 --> 01:16:35,060 But it does exist in closed form. 855 01:16:43,623 --> 01:16:45,040 If you're interested in this, read 856 01:16:45,040 --> 01:16:51,680 about what is called the Kerr-Newman solution. 857 01:16:51,680 --> 01:16:55,030 So if you're keeping score, we have this spherically symmetric 858 01:16:55,030 --> 01:16:58,030 black hole, which only has a mass, 859 01:16:58,030 --> 01:17:02,030 you have the charged black hole whose name I forgot to list. 860 01:17:02,030 --> 01:17:02,530 Ah! 861 01:17:02,530 --> 01:17:03,730 Sorry about that. 862 01:17:03,730 --> 01:17:21,843 This guy is known as the Reissner-Nordstrom black hole. 863 01:17:21,843 --> 01:17:23,510 One of those O's, I believe, is supposed 864 01:17:23,510 --> 01:17:24,677 to have a stroke through it. 865 01:17:24,677 --> 01:17:27,580 Those of you who speak Scandinavian languages 866 01:17:27,580 --> 01:17:31,310 can probably spell it and pronounce it better than I can. 867 01:17:31,310 --> 01:17:32,860 So we have Schwarzschild, which is 868 01:17:32,860 --> 01:17:37,300 only mass, Reissner-Nordstrom, which is mass in charge, 869 01:17:37,300 --> 01:17:41,310 Kerr, which is mass and spin, and Kerr-Newman, which 870 01:17:41,310 --> 01:17:44,110 is mass, spin, and charge. 871 01:17:44,110 --> 01:17:47,010 You might start thinking, all right, well, 872 01:17:47,010 --> 01:17:48,400 does this keep going? 873 01:17:48,400 --> 01:17:54,520 Do I have a solution for a black hole that's got, you know, 874 01:17:54,520 --> 01:17:58,080 northern hemisphere bigger than the southern hemisphere? 875 01:17:58,080 --> 01:18:00,100 You know, every time you think about adding 876 01:18:00,100 --> 01:18:02,230 a bit of extra sort of schmutz to this thing, 877 01:18:02,230 --> 01:18:05,200 do I need another solution? 878 01:18:05,200 --> 01:18:07,990 Well, let me describe a remarkable theorem. 879 01:18:16,770 --> 01:18:40,080 The only stationary spacetimes in 3 plus 1 dimensions with 880 01:18:40,080 --> 01:18:47,980 event horizons are the Kerr-Newman black holes-- 881 01:19:02,480 --> 01:19:19,520 completely parameterized by mass, spin, and charge. 882 01:19:23,270 --> 01:19:26,720 If you take the Kerr-Newman solution, you set charge to 0, 883 01:19:26,720 --> 01:19:27,700 you get Kerr. 884 01:19:27,700 --> 01:19:30,230 If you take Kerr and you set spin to 0, 885 01:19:30,230 --> 01:19:32,010 you get Schwarzschild. 886 01:19:32,010 --> 01:19:33,600 So the Kerr-Newman solution gives me 887 01:19:33,600 --> 01:19:36,860 something that includes these other ones as sort of a subset. 888 01:19:36,860 --> 01:19:39,540 And what this theorem states is that the only-- so 889 01:19:39,540 --> 01:19:42,240 stationary means time-independent. 890 01:19:59,070 --> 01:20:01,650 So in other words, the only spacetimes 891 01:20:01,650 --> 01:20:05,760 that are not dynamical, but that have event horizons, 892 01:20:05,760 --> 01:20:09,050 at least with three space and one time dimension, 893 01:20:09,050 --> 01:20:10,650 are the Kerr-Newman black holes. 894 01:20:10,650 --> 01:20:14,340 Once you know these, you have characterized all black holes 895 01:20:14,340 --> 01:20:15,850 you can care about. 896 01:20:15,850 --> 01:20:20,610 And in fact, in any astrophysical context, 897 01:20:20,610 --> 01:20:23,070 any macroscopic object with charge 898 01:20:23,070 --> 01:20:26,580 is rapidly neutralized by ambient plasma that just sort 899 01:20:26,580 --> 01:20:28,520 of fills all of space. 900 01:20:28,520 --> 01:20:33,840 And so this Kerr solution, in fact, 901 01:20:33,840 --> 01:20:38,610 gives an exact mathematical description to every black hole 902 01:20:38,610 --> 01:20:41,740 that we observe in the universe. 903 01:20:41,740 --> 01:20:44,840 That is an amazing statement. 904 01:20:44,840 --> 01:20:45,340 OK? 905 01:20:45,340 --> 01:20:48,430 Of course, as a physicist, you want to test this. 906 01:20:48,430 --> 01:20:50,710 And this, in fact, is much of what 907 01:20:50,710 --> 01:20:53,220 my research and research of many of my colleagues is about. 908 01:20:53,220 --> 01:20:56,470 Can we actually formulate tests of this Kerr hypothesis? 909 01:20:56,470 --> 01:20:58,210 And many of us have spent our careers 910 01:20:58,210 --> 01:21:00,010 coming up with such things. 911 01:21:00,010 --> 01:21:03,280 Suffice it to say, in the roughly negative 5 minutes 912 01:21:03,280 --> 01:21:06,850 I have left in this lecture, that the Kerr metric has 913 01:21:06,850 --> 01:21:09,310 survived every test that we have thrown at it. 914 01:21:12,660 --> 01:21:16,710 So this metric, like I said, was essentially 915 01:21:16,710 --> 01:21:21,000 derived by the mathematician Roy Kerr as his PhD thesis. 916 01:21:21,000 --> 01:21:26,880 And it has really earned him a place in physicist Valhalla. 917 01:21:31,245 --> 01:21:32,620 Let me just conclude this lecture 918 01:21:32,620 --> 01:21:33,910 by making one comment here. 919 01:21:39,105 --> 01:21:40,480 An important word in this theorem 920 01:21:40,480 --> 01:21:45,280 is the statement that the only stationary spacetimes 921 01:21:45,280 --> 01:21:47,320 are the Kerr-Newman ones, stationary spacetimes 922 01:21:47,320 --> 01:21:49,810 with event horizons, so the Kerr-Newman black holes. 923 01:21:49,810 --> 01:21:56,380 What this means is that when a black hole forms, 924 01:21:56,380 --> 01:22:02,920 it may be dynamical, it may not yet be stationary. 925 01:22:02,920 --> 01:22:06,610 And so the way that this theorem, which 926 01:22:06,610 --> 01:22:25,150 is known as the No-Hair theorem, the way that it is enforced 927 01:22:25,150 --> 01:22:31,240 is that, imagine I have some kind of an object that 928 01:22:31,240 --> 01:22:34,090 due to physics that we don't have time to go into here, 929 01:22:34,090 --> 01:22:38,260 imagine that this thing, its physics changes in such a way 930 01:22:38,260 --> 01:22:41,350 that its fluid can no longer support 931 01:22:41,350 --> 01:22:45,510 its own mass against gravity, and it 932 01:22:45,510 --> 01:22:47,680 collapses to a black hole. 933 01:22:47,680 --> 01:22:48,180 OK? 934 01:22:48,180 --> 01:22:51,375 Initially, this could be a huge, complicated mess. 935 01:22:57,960 --> 01:23:07,510 So we have a mass, charge, spin, magnetic fields, who knows? 936 01:23:15,790 --> 01:23:22,120 The No-Hair theorem guarantees that after some period of time, 937 01:23:22,120 --> 01:23:25,280 it will be totally characterized by three numbers-- the mass, 938 01:23:25,280 --> 01:23:28,470 the spin parameter a and the charge q. 939 01:23:28,470 --> 01:23:38,500 What goes on is that during the collapse process, 940 01:23:38,500 --> 01:23:40,208 radiation is generated. 941 01:23:43,560 --> 01:23:46,650 What this radiation does is it carries away 942 01:23:46,650 --> 01:23:49,890 gravitational waves, carries away electromagnetic waves. 943 01:23:49,890 --> 01:23:53,100 Some of this is actually absorbed by this black hole. 944 01:23:53,100 --> 01:23:58,080 And it does so in such a way that it precisely cancels out 945 01:23:58,080 --> 01:24:00,570 everything in the spacetime that does not 946 01:24:00,570 --> 01:24:03,210 fit the Kerr-Newman form. 947 01:24:03,210 --> 01:24:10,360 You wind up-- so this is one of these things where we really 948 01:24:10,360 --> 01:24:14,830 can only probe this either with observations that sort of look 949 01:24:14,830 --> 01:24:17,770 at things like black holes and compact bodies 950 01:24:17,770 --> 01:24:20,500 colliding and forming black holes 951 01:24:20,500 --> 01:24:22,540 and looking at what the end state looks like. 952 01:24:22,540 --> 01:24:24,640 Or we can do this with numerical experiments 953 01:24:24,640 --> 01:24:27,310 where we simulate very complicated collapsing 954 01:24:27,310 --> 01:24:29,440 or colliding objects on a supercomputer 955 01:24:29,440 --> 01:24:31,850 and look at what the end result is. 956 01:24:31,850 --> 01:24:35,470 And what we always find is that the complex radiation that 957 01:24:35,470 --> 01:24:38,860 is generated in the collapse and the collision process 958 01:24:38,860 --> 01:24:42,550 always shaves away every bit of structure, 959 01:24:42,550 --> 01:24:45,070 except for exactly what is left to leave it 960 01:24:45,070 --> 01:24:47,087 in the Kerr-Newman form at the end. 961 01:24:47,087 --> 01:24:48,670 Really, when we do these calculations, 962 01:24:48,670 --> 01:24:50,462 we generally wind up with a Kerr black hole 963 01:24:50,462 --> 01:24:52,840 because we tend to study astrophysical problems that 964 01:24:52,840 --> 01:24:54,820 are electrically neutral. 965 01:24:54,820 --> 01:24:57,760 So this is a result that is sometimes 966 01:24:57,760 --> 01:25:05,000 called Price's theorem, based on sort 967 01:25:05,000 --> 01:25:08,510 of foundational calculations that were done 968 01:25:08,510 --> 01:25:10,340 by my friend Richard Price. 969 01:25:10,340 --> 01:25:15,770 He did much of this right around the time I was born in the days 970 01:25:15,770 --> 01:25:19,700 of being a PhD student and looking at the behavior 971 01:25:19,700 --> 01:25:22,880 of highly distorted black holes and seeing how the No-Hair 972 01:25:22,880 --> 01:25:24,230 theorem-- 973 01:25:24,230 --> 01:25:25,940 You can imagine making a spacetime that 974 01:25:25,940 --> 01:25:27,320 contains what should be a black hole, 975 01:25:27,320 --> 01:25:28,445 but you somehow distort it. 976 01:25:28,445 --> 01:25:31,430 What you find is it becomes dynamical 977 01:25:31,430 --> 01:25:34,682 and it vibrates in such a way as to get rid of that distortion. 978 01:25:34,682 --> 01:25:36,140 And you leave behind something that 979 01:25:36,140 --> 01:25:40,600 is precisely Kerr or Kerr-Newman if you have charge. 980 01:25:40,600 --> 01:25:42,770 Price's theorem is a semi-facetious statement 981 01:25:42,770 --> 01:25:48,290 that tells me everything in the spacetime that can be radiated, 982 01:25:48,290 --> 01:25:50,030 is radiated. 983 01:25:50,030 --> 01:25:59,280 In other words, any bit of structure, any bit of structure 984 01:25:59,280 --> 01:26:01,500 in the spacetime that does not comport 985 01:26:01,500 --> 01:26:05,280 with the Kerr-Newman solution radiates away 986 01:26:05,280 --> 01:26:06,730 and only Kerr-Newman is left. 987 01:26:09,980 --> 01:26:12,340 So that concludes this lecture. 988 01:26:12,340 --> 01:26:17,050 My final lecture, which I will record in about 15 minutes, 989 01:26:17,050 --> 01:26:20,050 is one in which we are going to look 990 01:26:20,050 --> 01:26:22,480 at one of the ways in which we test this spacetime, which 991 01:26:22,480 --> 01:26:25,900 is by studying the behavior of orbits 992 01:26:25,900 --> 01:26:27,830 going around a black hole. 993 01:26:27,830 --> 01:26:30,210 So I will stop here.